We are up on the south outskirts of Leicester (Braunstone) at the weekend, travelling with the dogs, and I am trying to find a good place to let the dogs out mid journey for a good run.  As I don't know the area at all I was hoping someone might have some recommendations.

Not sure where you are coming from and whereabouts you wil be stopping but a few nice places within Leicestershire are:

Watermead Country Park, Syston

Foxton Locks

Rutland Water

Beacon Hill Country Park

Bradgate Park, Newtown Linford

Burrough Hill, Melton

http://www.leics.gov.uk/index/community/community_services/environment_and_heritage/country_parks/country_parks_market_bosworth_park.htm

You should be able to find details of all these on the internet.

I hope this helps!