I took our two out to the beach last night for a Valentines walk with the wife (we have a dachshund and a retreiver - both under 2 so quite a handful!) and I covered the seat too but they just went mad when they got back in the car and it was not so much the dirt but the smell of whatever the heck they were rolling/walking/running thru or in! lol
Had the windows down most of today to get rid of the odour.