If you ever look or read about the old spy tactics, they would pass messages in soda cans under a park bench, or do things that couldn't easily be found. From a traffic analysis and meta-data collection issue, encryption of the messages do not matter since the analysis is of the signals not the content. Signals analysis may look at things like between what two parties, frequency, etc.
One possibility may be using something like a TOR-like infrastructure, where the traffic is difficult to decipher without controlling a lot of nodes, though as recent news has shown, TOR is not a full proof way to get anonymity and can be compromised.
There are also anonymous BBS and BBS-type systems. You post the message encrypted and then someone else has to know where to go to find it an decrypt it. If you upload and download a lot, its difficult to tell the connections between individuals, making traffic analysis hard.
Focusing on meta-data or signals analysis part, you can always look at adding noise, such as the Chaffing and Winnowing system proposed by Ronald Rivest. You could also try to communicate over covert channels, split messages, etc. You could move around between various forums every few days, even if they are off topic (and thus less likely to be targeted in a search). You could communicate in the comments of random websites and have some way of telling your receiver how to find the comments. You could also implement stenography and post things in very public places.
In general, to make it easy and secure would be real challenge. If all traffic can be monitored at the wire level, then traffic analysis is always possible without sufficient noise or physically moving between networks (go to a new coffee shop every few minutes).
If you are not concerned about the meta data and the signals, but want privacy, any type of encryption where the service provider does not hold the keys will work. Ekiga and other tools provide encryption, and are relatively user friendly. Even outside of security, there are always arguments for replacing email, its an issue of what features are needed by the masses and how to get them to transition. The Internet is full of legacy hold backs which take a long time to be replaced (even WWW-browsing has lots of limitations and draw backs).
Just saw this today on HN: Secure Share. Some of their points:
We employ GNUnet for peer-to-peer routing and encryption (the new
openssl of P2P) and PSYC to create the social trust graph (because it
performs a dozen times better than XMPP or OStatus).
communication between parties cannot be measured as they may have none to several routing hops in-between. an observer never knows if a communication came where it came from and ends where it is going to.
May end up being something useful, if you could get people to agree to leave the web browser.