What I do is check the plants once or twice a week, particularly the buds and the underside of young leaves, and simply squish any I find.
If you do this on a regular basis, they never build up sufficient numbers to become a problem - any that do escape my thumb and finger end up on the ladybird's dinner menu!
Blackspot can't be fully eradicated, even with chemicals, as the spores always survive in the soil etc. Easiest thing is to pick off the affected leaves, making sure you remove any that have fallen off the bushes also, and put these in the bin - not on the compost heap.
Mildew thrives in poorly ventilated situations - make sure there is enough air getting into and around your rose bushes - prune them so as to try and have the bush growing in a 'goblet' shape, with the centre clear of any growth. Also, bushes stressed by dry conditions tend to be very susceptible to mildew, so give them a good watering weekly in very dry weather (one long soaking is infinitely better than daily sprinkling).
Otherwise, there are plenty of modern roses which are beautiful, have excellent scent, and have been bred with some resistance to blackspot and mildew. If you're fighting a losing battle, sometimes you're best off getting rid of particularly disease susceptible plants and either buying disease resistant varieties, or not growing roses at all (a la the late great Christopher Lloyd).