Changing sign of elements of a list on even positions

I have a list of lists of equal sizes (odd or even), e.g.

list={{1,-2,3,1,3,4},{3,5,6,-1,9,0},{1,-2,-4,-5,1,5},{1,-1,2,3,5,4},...} 

I would like to change signs of every element which is on even or odd position. In the example above, for the choice of change on even positions, it should give the following result:

 listnew={{1,2,3,-1,3,-4},{3,-5,6,1,9,0},{1,2,-4,5,1,-5},{1,1,2,-3,5,-4},...} 

What is the most effective way to achieve this? (The length of list may be huge here).