It is so much harder than it was when I started. More and more artists want to write thier own material, people don't accept unsolicited material for fear of lawsuits and the days of the Brill building contract writers are mostly a distant historical footnote.
However, two things still hold true - a good song is a good song and perserverance pays.
If you really have the goods, and if you pay your dues (contact publishers, producers, artists, etc. etc) you can still place a song. You may have to accept plenty of corrective criticism and deal with plenty of rejects before you find someone willing to give you a shot, but it can happen. Shameless self promotion and aggressive networking are an absolute must.
I've have some "success" using the Songwriter's Market and through a Taxi membership. Success meaning getting some material placed with meager checks coming in - (I don't even imagine success on the level of a Diane Warren) However, I've suffered more rejection than the human ego should, and have compiled some pretty hefty postage/phone bills over the years - but it's part of the process.
As Aaron indicates, the days of a piano/vocal demo are long gone. You need a well produced demo, with a great vocal performance. Good demo singers can cost $100 to $200 per song (this does not include the other production costs). While you can find good singers for less, the people who can ace it in a few takes can be well worth the extra money.
The most successful local writer I know is also the most focused, dogmatic person I've ever met. He is a clever wordsmith who writes decent melodies over rather generic chord progressions - but - he is a member of every songwriter association known to man. He approaches every local band, producer and studio trying to get people to perform/record his songs. He spends his vacation every year (yeh - he still needs a day job) on a trip to Nashville (planned in advance, with appointments already made, etc).
He has had quite a few songs publish/placed/recorded and gets small checks on a consistant basis. I truly believe he will eventually get a song on a high charting recording - not because he is a "great writer" (he is a "good writer"), by mainly because he keeps at it year after year.