This phase I/II trial studies the side effects and best dose of temsirolimus when given together with cladribine and rituximab and to see how well it works in treating patients with newly diagnosed mantle cell lymphoma. Monoclonal antibodies, such as rituximab, may interfere with the ability of cancer cells to grow and spread. Drugs used in chemotherapy, such as cladribine, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Temsirolimus may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Giving temsirolimus together with cladribine and rituximab may kill more cancer cells.