Lenalidomide and Dexamethasone With or Without Thalidomide in Treating Patients With Multiple Myeloma

Inclusion Criteria: * Patients must be diagnosed with symptomatic multiple myeloma within the past 90 days confirmed by the following: * Bone marrow plasmacytosis with \>= 10% plasma cells or sheets of plasma cells or biopsy proven plasmacytoma which must be obtained within 4 weeks prior to randomization * Measurable levels of monoclonal protein (M protein): \>= 1.0 g/dL on serum protein electrophoresis or \>= 200 mg of monoclonal light chain on a 24 hour urine protein electrophoresis which must be obtained within 4 weeks prior to randomization; both serum protein electrophoresis (SPEP) and urine protein electrophoresis (UPEP) are required to be performed within 28 days prior to randomization; please note that if both serum and urine m-components are present, both must be followed in order to evaluate response * Hemoglobin \> 7 g/dL * Platelet count \> 75,000 cells/mm\^3 * Absolute neutrophil count \> 1000 cells/mm\^3 * Creatinine \< 2.5 mg/dL and creatinine clearance (measured or calculated) \>= 60 mL/min * Bilirubin =\< 1.5 mg/dL * Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) and serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) =\< 2.5 times the upper limit of normal * Prior palliative and/or localized radiation therapy is permitted provided at least 4 weeks have passed from date of last radiation therapy to date of registration; patients with prior solitary plasmacytoma treated with radiation therapy with curative intent are eligible if the disease has now progressed to active multiple myeloma meeting all the eligibility criteria for this protocol * Eastern Cooperative Oncology Group (ECOG) performance status 0, 1, or 2 * Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 25 mIU/mL within 10-14 days and again within 24 hours prior to starting cycle 1 of lenalidomide; further, they must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control: one highly effective method (intrauterine device \[IUD\], birth control pills, tubal ligation or partner's vasectomy) and one additional effective method (condom, diaphragm or cervical cap); FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP, even if they have had a successful vasectomy starting 4 weeks prior to and while taking CC5013 or thalidomide and for four weeks after discontinuing this therapy; a FCBP is a sexually mature woman who: has not undergone a hysterectomy or bilateral oophorectomy; or has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months); all patients must be counseled by a trained counselor every 28 days about pregnancy precautions and risks of fetal exposure * Patients with a history of prior malignancy are eligible provided there is no active malignancy and a low expectation of recurrence within 6 months Exclusion Criteria: * No prior systemic therapy with the exception of bisphosphonates for multiple myeloma * Prior glucocorticosteroid therapy for the treatment of multiple myeloma is not permitted; prior systemic glucocorticosteroid use for the treatment of non-malignant disorders is permitted; concurrent use after registration on the study should be restricted to the equivalent of prednisone 10 mg per day; prior or concurrent topical or localized glucocorticosteroid therapy to treat non-malignant comorbid disorders is permitted * Patients must not have active, uncontrolled seizure disorder; patients must have had no seizures in the last 6 months * Patients must not have uncontrolled intercurrent illness including uncontrolled hypertension, symptomatic congestive heart failure, unstable angina, uncontrolled cardiac arrhythmia, uncontrolled psychiatric illness or social situation that would limit compliance with the study, or a prior history of Stevens Johnson syndrome * Patients with smoldering myeloma or monoclonal gammopathy of undetermined significance are not eligible * Patients must not have grade 2 or higher peripheral neuropathy due to other medical conditions at the time of randomization * Patients must not have active, uncontrolled infection * Patients must not have a history of current or previous deep vein thrombosis or pulmonary embolism regardless of whether or not the patient is receiving anticoagulation therapy * For patients registered prior to activation of Addendum # 6; patients must be willing and able to take prophylaxis with either aspirin at 325 mg/day or alternative prophylaxis with either low molecular weight heparin or Coumadin * For patients registered after activation of Addendum # 6; patients entering the expansion phase of the protocol, which tests anticoagulant prophylaxis, must be able and willing to be randomized between aspirin at 325 mg/day and Coumadin * Female patients MUST NOT be pregnant or breastfeeding; due to the potential teratogenic properties of CC 5013, and the known teratogenicity associated with thalidomide, the use of these drugs in this patient population is ABSOLUTELY CONTRAINDICATED