Ixazomib Citrate and Lenalidomide After Stem Cell Transplant in Treating Patients With Newly Diagnosed Multiple Myeloma

Inclusion Criteria:

* Patient must have undergone autologous stem cell transplantation, with melphalan as a preparative regimen, within 12 months of initiation of induction therapy for newly diagnosed myeloma
* Time to initiation of maintenance therapy; patients may start maintenance therapy as early as 60 days post-transplant and up to 180 days post-transplant; as long as they meet the following criteria:
* Platelet count \>= 100,000/mm\^3; platelet transfusions to help patients meet eligibility criteria are not allowed within 3 days before study enrollment
* Neutrophil count \>= 1000/mm\^3; (no growth factors within 5 days prior to first dose of the study drug)
* Total bilirubin =\< 1.5 x upper limit of normal (ULN)
* Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN
* Creatinine \< 2.5 mg/dL
* Recovered (i.e., =\< grade 1 toxicity) from the reversible effects of autologous stem cell transplant
* Patients whose primary therapy was changed due to suboptimal response of toxicity will be eligible, however no more than 2 regimens will be allowed prior to ASCT
* Patients must have an Eastern Cooperative Oncology Group (ECOG) status of 0 to 2
* Voluntary written informed consent before performance of any study-related procedure not part of normal medical care, with the understanding that consent may be withdrawn by the subject at any time without prejudice to future medical care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 childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent, during study treatment and for 90 days after the last dose of study treatment, AND

  * Must also adhere to guidelines of any treatment-specific pregnancy prevention program, if applicable, 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 treatment, OR

  * Must also adhere to guidelines of any treatment-specific pregnancy prevention program, if applicable, 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)

Exclusion Criteria:

* Patient has \>= grade 3 peripheral neuropathy, or grade 2 with pain on clinical examination during the screening period
* Major surgery within 14 days before the first dose of study drug
* 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 MLN9708
* Known active central nervous system involvement
* Systemic treatment, within 14 days before study enrollment, with strong inhibitors of CYP1A2 (fluvoxamine, enoxacin, ciprofloxacin), 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
* Inability to swallow oral medication, inability or unwillingness to comply with the drug administration requirements, or gastrointestinal (GI) procedure that could interfere with the oral absorption or tolerance of treatment
* 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
* Female subject who are lactating or have a positive serum pregnancy test during the screening period
* Serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with participation or completion of treatment according to this protocol
* Corrected QT interval using Bazett's formula (QTcB) \> 470 milliseconds (msec) on a 12-lead electrocardiogram (ECG) obtained during the screening period; if a machine reading is above this value, the ECG should be reviewed by a qualified reader and confirmed on a subsequent ECG
* Ongoing or active systemic infection, known human immunodeficiency virus (HIV) positive, known active hepatitis B virus hepatitis, or known active hepatitis C virus hepatitis
* Infection requiring systemic antibiotic therapy or other serious infection within 14 days before study enrollment
* Known allergy to any of the study medications, their analogues, or excipients in the various formulations
* Co-morbid systemic illnesses or other severe concurrent disease that, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens
* 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