For the past few years that I've had a cell phone, I've been known to be very bad at checking my voicemails. It's so clumsy, and if not checked soon enough, they pile and pile. If someone calls and there is a voicemail that you just must listen to, you have to go through ALL previous voicemails first. This is the same way voicemail was checked
years ago, and I think it's time for a change. I'm glad others agree with me.
There are now services that can take over as your voicemail service. When you miss someone's call, the caller is redirected to your new voicemail service instead of your service provider's.
For a while, I was using Privatephone.com. PrivatePhone gives you your own local phone number that is intended for use as a personal number that you give out for spam-style offers online, or to people you meet randomly that ask for your number while you really don't want to give it out. I found a useful link at Howard Forums that details how to enable call forwarding on T-Mobile (my provider) phones, and I went ahead and gave it a shot. Months later, I was satisfied with my decision.
With PrivatePhone, I was able to check my voicemails through my phone like before, through email, or through a web-based email-like interface. I had .wav of each voicemail emailed to me, so I was notified immediately by Thunderbird as soon as an email arrived. In addition, PrivatePhone sent me a text message telling me a voicemail was left, with the caller's number. The text message only showed the caller's number (along with a short PrivatePhone message), even if the caller was in my PrivatePhone addressbook.
I then stumbled upon YouMail. YouMail did everything PrivatePhone failed to do. It not only is a third-party voicemail service, but it also has excellent features such as custom voicemail greetings for each caller, text messages with the caller's number, name (if the caller is not in your contact list, it shows the caller ID name), and length of the voicemail. In addition, the emailed copies of the voicemails were sent in .mp3 as opposed to the .wav of PrivatePhone; I had bad luck with VLC and .wav files for some reason. The only drawbacks to YouMail are that you don't get your own private number (everyone uses the same voicemail number; your voicemail is handled based on the number that forwards the call to them), and that you can't currently set caller groups. It would be nice to have friends, family, work, etc., be greeted with different voicemail messages, but not have to individually set the message for each individual caller. Still, this service to me far exceeds what PrivatePhone could offer me, and I am still using it today.
If you have multiple phones or need a number you can comfortably give out to near-strangers, you might want to try GrandCentral. If a caller dials your GrandCentral number, GrandCentral can have multiple phones ring at the same time. Think of this as phone redundancy when you have service problems with your phone(s). GrandCentral used to offer ringback tones (where callers are greeted by music or whatever you decide to upload), but since being bought by Google, that feature has disappeared (for copyright reasons I'm sure). Still, there is a library of ringback tones you can choose, but none of them are anything special. I suggest you give this a shot, I'm sure you'll like it 
UPDATE: YouMail has recently made tons of changes to the site, and it appears to be getting better every week. I strongly advise you check it out. Whenever I think of a possible improvement, I end up seeing it implemented shortly after. I'm quite impressed with the service. Let me know what you think about it in the comments.
UPDATE: Privatephone.com will be ending their service on December 31, 2007. FreeDigits was the closest replacement I could find to Privatephone, but they now no longer offer dedicated phone numbers; they instead make you use a "short code" which is the same as an extension. My guess is that they didn't want to pay for more phone numbers. If you're still interested in that service, they offer it through a sister site, ringtonumber.com.