I have no time for F. It hurts my hand and is can't be relied upon to produce a nice sound. So here's what I do:

See that fat, low E string that's right under your chin? If you put your finger on the first fret, it plays the note F.

The string right below is an A. Now you have two of the notes of the F chord. Since in a song in C, the F chord usually occurs for only two beats at a time, it's easy to take those two notes and fake something.

One day I was doing that, and our harper complimented me on my cello sound.
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Although, for various reasons, when I get a good tune in C, I usually change it to D, and everybody's happier. That gets rid of the F's AND the C's.