Hi
I'm guessing that you must be lower than your neighbours if you get all the run-off?
It may be that you have a cultivation or iron pan which is stopping the water draining away - these can be broken up or removed (if iron pan) but you may have to dig down quite deep to find it - might be worth digging a test pit in the worst area to see if that is the case.
You don't say what sort of soil you have - if high clay content drainage can be a problem but can be improved over time, by digging in lots of organic matter (compost and manure) and also liming. Liming causes something called flocculation which is plain english means it starts to encourage the soil to develop larger particles which will improve the structure giving more air pores for drainage - you should be able to get a suitable product from your garden centre. Don't go on the soil or cultivate it when wet as that will make the problem worse. If it's loam/sandy then I suspect it may well be a pan blocking the waters path down.
Good luck!