When 5.7.5 is released ( you can try out the release candidate from the downloads page) you'll be able to do this from the Edit Block Design command, from the Device & Visibility section of the effects palette. See the attached image. (Just uncheck the devices that you don't want to show the block on.)

Note: You're theme will have to make use of a grid framework that supports this. Currently that means Bootstrap and Foundation.

Interesting concept Andrew, 'Device & Visibility', will look at that.

@Jim, you can also use @media queries to hide what ever you want from mobile devices.

Thanks Andrew I'm looking forward to seeing that features in next update.

SteevB I have a feeling @media queries is a CSS thing?! Since I really don't know CSS do you think it something that I'm capable doing whilst I await the next release of C5?

Thanks
Jim

Worked perfectly - thanks for posting this.