I use strips. But because I like the variety of color placement and extreme scrappiness, I will sometimes cut strips shorter to help avoid the dread repetition of fabric and color in the same place on blocks.
I've found that it's not necessary to be uptight about absolute square construction, particularly if your logs are fairly narrow (my preferred width is 1.5 inches, thought I will go as wide as 2.25). Once the quilt is constructed, small variations in block size and shape are not an issue, and actually add to my enjoyment of the patterns.
I would never have the patience to foundation piece a log cabin block. That sounds like torture to me.
RST