Pemetrexed and Carboplatin in Treating Patients With Recurrent Ovarian, Primary Peritoneal, or Fallopian Tube Cancer

Inclusion Criteria: * Patients must have a histopathologically confirmed diagnosis of epithelial ovarian, primary peritoneal, or fallopian tube carcinoma * Patients must have received at least 1 prior platinum and taxane based chemotherapy regimen; patients may have failed no more than 2 prior chemotherapy regimens * Patients must have "platinum sensitive" disease, which will be defined as those patients with relapsed disease who had an initial complete remission, and relapsed more than 6 months after completion of initial platinum based chemotherapy * Recurrent disease must be confirmed by: * Bidimensionally measurable disease which can be measured by physical examination or by means of medical imaging techniques (measurable disease) * Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest dimension to be recorded); each lesion must be \>= 2.0 cm when measured by conventional techniques, including palpation, x-ray, computed tomography (CT), and magnetic resonance imaging (MRI), or \>= 1.0 cm when measured by spiral CT; all measurable lesions up to a maximum of 5 lesions per organ and 10 lesions in total representative of all involved organs should be identified as target lesions and will be recorded and measured at baseline; all baseline evaluations of disease status should be performed as close as possible to the start of treatment and never more than 4 weeks before the beginning of treatment * Target lesions should be selected on the basis of their size (lesions with the longest dimension, LD) and their suitability for accurate repetitive measurements by one consistent method of assessment (either clinically or by imaging techniques); a sum of LD for all target lesions will be calculated and reported as the baseline sum LD; the baseline sum LD will be used as reference to further characterize the objective tumor response of the measurable dimension of the disease; all other lesions (or sites of disease) should be identified as non-target lesions and should also be recorded at baseline OR * Two confirmed serum cancer antigen-125 (CA-125) levels greater than or equal to 70 u/ml (or 2 x upper limit of normal) separated by 1 week and obtained within 4 weeks prior to entry to the study (evaluable disease) * Patients must not have had other myelosuppressive therapy within four weeks of initiating pemetrexed/ carboplatin therapy * Patients must have recovered from effects of recent surgery * Patients must have a Gynecologic Oncology Group (GOG) performance status of 0, 1, or 2 * White blood cell (WBC) greater than or equal to 3,000/ul * Platelet count greater or equal to 100,000/ul * Neutrophil count greater or equal to 1,500/ul * Creatinine clearance \>= 45 ml/min (estimated creatinine clearance by Cockcroft-Gault equation acceptable) * Total bilirubin =\< to 1.5 mg/dL * Serum glutamic oxaloacetic transaminase (SGOT) and serum glutamate pyruvate transaminase (SGPT) =\< three times the upper normal institutional limits; if patient has known hepatic metastases, patients may be enrolled if liver function test is =\< five times the upper normal institutional limits * Alkaline phosphatase =\< three times the upper normal institutional limits; if patient has known hepatic metastases, patients may be enrolled if liver function test is =\< five times the upper normal institutional limits * Patient must have signed informed consent * Patients must be willing to take the dexamethasone, folic acid and vitamin B12 supplementation as indicated in the protocol to reduce adverse drug toxicity * Patients must be willing to interrupt aspirin and other nonsteroidal anti-inflammatory drugs (NSAID) intake for 2 days before, day of, and 2 days after each chemotherapy treatment; low dose 80 mg aspirin and cyclooxygenase-2 (Cox-2) inhibitors are excluded from this restriction; if concomitant administration of an NSAID is necessary, patients should be monitored closely * Patients must have a life expectancy of greater than 12 weeks * Patients may not have concurrent or previous invasive malignancies, with the exception of non-melanoma skin cancer or no evidence of recurrence of previous malignancy within the last 5 years * Patients must have a current exam, blood work and any clinically indicated imaging studies within 4 weeks prior to study enrollment * Baseline folate and homocysteine blood levels * The ability to interrupt NSAIDS 2 days before (5 days for long-acting NSAIDs), the day of, and 2 days following administration of Alimta * The ability to take folic acid, vitamin B12, and dexamethasone according to protocol Exclusion Criteria: * Patients who have had more than two prior chemotherapeutic regimens * Patients who have had prior treatment with pemetrexed * Patients with a GOG performance status of 3 or 4 * Patients with \>= grade 2 neuropathy * Patients who have received external beam whole pelvic or whole abdominal radiation treatment (\>= 4500 centigray \[cGy\]) which would limit vascular capacity and reduce adequate drug delivery * Patients with evidence of recurrence from another malignancy within the previous five years * Patients with a concomitant malignancy other than squamous cell skin cancer * Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, unstable angina pectoris, or psychiatric illness/social situations that would limit compliance with study requirements * Patients who have received an investigational drug within the last 30 days that has not received regulatory approval * Presence of third space fluid which cannot be controlled by drainage; for patients who develop or have baseline clinically significant pleural or peritoneal effusions (on the basis of symptoms or clinical examination) before or during initiation of Alimta therapy, consideration should be given to draining the effusion prior to dosing; however, if, in the investigator's opinion, the effusion represents progression of disease, the patient should be discontinued from study therapy