In my last three blood test, my platelets count was near 120. My doctor told me to not worry about it. Could it be a sign of leukemia?

See below... Theoretically, significant thrombocytopenia (low platelets) is one of the many abnormalities that can be associated with leukemia; however, many other conditions can cause it. In your case, the platelet number is only slightly decreased (normal range: 150-450 x 10 to the ninth/l) and, in the absence of other signs and symptoms, it is probably not indicative of leukemia.
Not likely. There are many causes for a low platelet count, and 120, 000 is only minimally reduced. Leukemia would be suspected if there was more wrong with the blood counts than just the platelets, and all values were getting progressively worse. Minimally low platelets can be due to clumping of platelets, in essence a laboratory error. If truly low, most common cause is immunologic (itp).