Round the side where all the cables are shoved is a tie wrap, try feeding the cables through that and tightening up a little.
Also if you could cut out 2 holes, 1 at the bottom and 1 at the top it would make a massive differance as your cables could be threaded through the bottom hole near the psu and reappear at the top near the atx connection.
Otherwise some simple suggestions are to get an 8 pin extension and some fan extension cables...Yes this means more cables but it also gives you something to play with enabling you to thread and feed the cables around the mobo rather than across it.
I have also found with the Antec 300 there are quite a few little metal connection clips, get some small tie wraps or zip ties and just secure any cables that run near them so they aren't hanging.
Final suggestion, if you go as far as cutting out holes spray the inside black and put a side window on.
that case looks similar to my Gf's case, check my links for some cable stupidity third link is her pc
also 3/10, mainly because you now are aware of C.M. and wish to do something about it
morning after edit*
turn your hard drives around 180' and have their sata/power leads fed in round the front of the case,
find some way of feeding the 8pin cpu either behind,around the mobo or down the side of the psu, up the wall (in the corner where that black switch is?) over the fan and tuck it so you just have enough free to reach the cpu socket
all those loose fan wires flailing about, either feed them someway out of sight or tape them together so it is just one black wire not spaghetti, same with the multicoloured spray of wires up by the heatsink
also tape the loose connector from audio back onto the main cable so it cant fall into the psu
wtf is that black switch thing anyway? lol
failing all that, bring it to me
I think a good approach is to cut as few holes as possible,
but always be willing to do so if required
there is a definite buzz from feeding cables through tiny recesses like the rolled section of a drivecage, but sometimes you just gotz to git you chop on hehe
Most of the cables are shoved behind, in the little window on the other side of the case. Are there any other improvements or things I could do?
For some of the fan cables, whenever i try to use something to tie it up to parts of the case (to hide it), it wont reach the window on the other side of the case.
I don't really wish to drill any holes or mod it in anyway.
Seems like the antec 300 is a tricky case to manage cables in.
I recently switched to a case with a bottom mounted psu(coolermaster elite 430) and had similar trouble with cable management luckily theres a small gap between the side of the case and the side panel and i managed to squeeze a few in there.
one thing i dont like about bottom mounted psu cases is you have to have wires going across the whole board which is something i didn't really encounter with top mounted.
think my case is a bit longer than your's not sure, but i managed to tuck a good few cables between the side panel and the hard drive bays. if i can get a clear picture of it i will post it.