Alternating Ixazomib Citrate and Lenalidomide as Maintenance Therapy After Stem Cell Transplant in Treating Patients With Multiple Myeloma

Inclusion Criteria: * Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status 0, 1, or 2 * Any autologous patient who underwent high dose melphalan (\>= 140 mg/m\^2) therapy/peripheral blood stem cell (PBSC) rescue for any stage of multiple myeloma and did not participate in another clinical transplant trial whose primary endpoint is also evaluating long-term, disease-free survival or survival; consenting for study between 30 days to 120 days after transplant; earliest can start therapy is 30 days post transplant after recovered from acute toxicity of autologous stem cell transplant (ASCT) * Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care * Female patients who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent form through 90 days after the last dose of study drug, OR * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject; (periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception) * Male patients, even if surgically sterilized (i.e., status post-vasectomy), must agree to one of the following: * Agree to practice effective barrier contraception during the entire study treatment period and through 90 days after the last dose of study drug, OR * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject; (periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception) * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 * Platelet count (transfusion independent) \>= 75,000/mm\^3 * Total bilirubin =\< 1.5 x the upper limit of the normal range (ULN) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN * Calculated creatinine clearance \>= 30 mL/min Exclusion Criteria: * Female patients who are lactating or have a positive serum pregnancy test during the screening period * Failure to have fully recovered (i.e., =\< grade 1 toxicity) from the reversible effects of prior ASCT chemotherapy * Major surgery within 14 days before enrollment * Radiotherapy within 14 days before enrollment; if the involved field is small, 7 days will be considered a sufficient interval between treatment and administration of the ixazomib * History of central nervous system multiple myeloma involvement * Infection requiring systemic antibiotic therapy or other serious infection within 14 days before study enrollment * Evidence of current uncontrolled cardiovascular conditions, including uncontrolled hypertension, uncontrolled cardiac arrhythmias, symptomatic congestive heart failure, unstable angina, or myocardial infarction within the past 6 months * Systemic treatment, within 14 days before the first dose of ixazomib, with strong inhibitors of cytochrome P450, family 1, subfamily A, polypeptide 2 gene (CYP1A2) (fluvoxamine, enoxacin, ciprofloxacin), strong inhibitors of cytochrome P450, family 3, subfamily A gene locus (CYP3A) (clarithromycin, telithromycin, itraconazole, voriconazole, ketoconazole, nefazodone, posaconazole) or strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of Ginkgo biloba or St. John's wort * Ongoing or active systemic infection, active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol * Known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent; patient cannot be allergic to boron * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib including difficulty swallowing * Diagnosed or treated for another malignancy within 2 years before study enrollment or previously diagnosed with another malignancy and have any evidence of residual disease; patients with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection * Patient has \>= grade 2 peripheral neuropathy, or grade 1 with pain on clinical examination during the screening period * Participation in other clinical trials, including those with other investigational agents not included in this trial, within 30 days of the start of this trial and throughout the duration of this trial * Patients with history prior to transplant of progression on lenalidomide therapy