I have an IBM ThinkPad z61t and the audio cuts in and out almost on a monthly basis - it'll work for about a month then not work for about a month. I've tried multiple operating systems and figured out that drivers aren't the problem - when it works it works on all of them and when it doesn't it doesn't. I know its a hardware issue, but I can't see anything wrong with it when I open it up and look at it. Help please.
That does sound like your audio chip has some sort of faulty connection. One thing you could try is to locate it on the motherboard and use a hair dryer or heat gun to reflow the connections. Stop blowing hot air just after seeing some solder melt. Also, make sure to cover any components you're not trying to fix with aluminium foil, because you might accidentally break a connection or damage a component. The challenge is to direct enough (and not too much) heat towards the chip. Also, you'd should find the location of the chip on the internet before trying.
If the above solution does not work, you can try reflowing the whole motherboard in the oven. I've seen reports of people putting the whole mainboard in the oven at 190 degrees for a similar repair, yielding mostly positive results. This is much more work, as you have to remove the CPU, CMOS battery, any tape and similar, thermal pads, wires and antennas etc. Find a description of how to do it here , or google "how to reflow a motherboard".
If neither solution works, getting a new mobo is the only solution that will get things back to how they were. A second hand motherboard can typically be found for 30-60 euro's for a similar machine, but availability might be an issue. Count 20 minutes to 6 hours labour depending on your experience (it takes me 20 minutes now, but 6 hours the first time, being very careful).
If you don't care about the microphone so much, you could buy a cheap USB external soundcard. There are options out there for <30 euro's which sound pretty decent (most likely better than the built in one). That can also have RCA output should you want to connect to an amplifier. If you have a bit more money and do care about a microphone you could get a similar USB interface, but one that includes a mic input.