Is there another way to find out if you have leukemia other than a bone marrow biopsy?

Yes. Often a bone marrow biopsy is needed to fully characterize a leukemia, but leukemia may be noted in the peripheral blood when it is smeared on a glass slide (pbs). Also, special tests on blood such as flow cytometry can (at least partially) characterize certain leukemias (eg, cll).
Peripheral Blood. Oftentimes leukemia can be diagnosed by just studying the peripheral blood drawn out of the veins. Sending the blood for a test called flow cytometry has made this possible. The value of doing a bone marrow aspiration and biopsy is to be able to grow cells for studying chromosome abnormalities that may help determine prognosis and the direction of treatment (such as timing of stem cell transplant).