Doing a Google search I found http://arxiv.org/vc/arxiv/papers/0901/0901.1857v1.pdf, which claims that "we prove that each integer-sided triangle can have at most two medians of integer length". However http://en.wikipedia.org/wiki/Integer_triangle states "There exist integer triangles with three rational medians.[9]:p. 64 The smallest has sides (68, 85, 87). Others include (127, 131, 158), (113, 243, 290), (145, 207, 328) and (327, 386, 409)."  An integral-sided triangle with rational medians can be changed into one with integral medians by multiplying all the sides by the LCM of the denominators of the rational lengths involved.

I haven't tried this multiplication on the other triangles mentioned in the Integer Triangle article, but assume that they do indeed have rational medians, which can be scaled up to integer medians.  They are not in the same generic solution, as the ratio of largest side to smallest side in the first is 174/136 ~=  1.279411764705882, and in the next one is 158/127 ~=   1.244094488188976. In (113, 243, 290) it's 290/113 ~=  2.56637168141593. With appropriate multiplication of the sides, the rational medians can be changed to integral ones.