In reply to gneiss boots:
My experience:
I let out my house through an agency. When something needs done they always e-mail me to ask if they can go ahead and do it essentially. Out of maybe 6 or 7 things so far, I've taken responsibility for every one except one. I arrange my own gas checks with a guy I've used before and is very reasonable - his rates are considerably less than the agency quoted for the job, I've done some DIY stuff myself, I've arranged for delivery and installation of new appliances myself and I've chased up contractors whose job developed faults after completion myself.
The only occasion I just told the agency to get on with it because I was abroad, I ended up with two bills from different companies for the same job, the first of which was basically "we couldn't fix this, you need a more serious contractor with more serious equipment" and the second more serious contractor was, predictably, quite expensive. I was pretty hacked off at the time but who can say if I would have managed to get it done any cheaper myself.
Everything leaves me with the bottom line feeling that getting the letting agency to get contractors to do something is exactly like getting the government to get contractors to do something: The middle man, who isn't really paying the money themselves, has little motivation to look for a good deal and every motivation to have a standard operating procedure which makes sure things go smoothly and don't become a debacle (while obviously notionally searching for competitive rates for a job so they can justify the expenditure).
So you have the choice: Leave it to the agency and accept that you will pay more for the privilege or take it on your own shoulders, save some money but know that that will be at the expense of a certain amount of your time and worry.
Which rather begs the question "wait, what am I paying an agency for then?!" to which my answer would be:
1) To find tenants, vet them, collect the rent, inspect the house on occasion and filter out spurious complaints while passing genuine ones on to you, the owner.
2) Insurance and insulation against non-paying/squatting/otherwise disaster tenant situations.
With number 2 basically being THE reason for me - if my tenants turn out to have been imported directly from hell, I should be completely insulated; my rent will continue to be paid even if the tenants are not doing so, legal costs covered, agency does all the legwork in progressing the reclamation of the property and house insurance reimburses any big repairs if they trash the place.