Reg Lindsay recorded it, but he didn't write it.
Copied from http://www.luma-electronic.cz/lp/d/Dean/dean_everybodysfav.htm

TEARS ON MY PILLOW
(Gene Autry - Fred Rose)
« © '41 Western Music »

We were so happy when love lingered on,
Back in our sweet used-to-be;
But now it's all over and you are gone,
Leaving a sad memory.

And there's tears on my pillow each morning.
I cry when I dream about you.
When I should be sleeping, I just lay there weeping,
Wond'ring if you're weeping too.

I'm trying so hard to forget you,
But that's not so easy to do,
With these tears on my pillow each morning,
Tears that I've shed over you.

[As sung by Jimmy Dean on "Everybody's Favorite" (Columbia CS 8827) 1963. Also Also recorded by Gene Autry, Jimmie Davis, Tompall Glaser, Kate Smith, and Bob Wills.]