BibleGateway, part of the Gospel Communications family of sites, keeps expanding its Bible Study tools and Resources. It now offers audio versions of the Bible. Current English language versions are the CEV, ESV, KJV, NASB, NIV, and TNIV. Currently the CEV is only available for New Testament readings.
For most Bibles, you can choose between a Flash Player or the Real Audio Player. I prefer the Flash Player, as it's a bit faster to start.
You may want to try listening to the Contemporary English Version (CEV) for New Testament passages. The CEV was specifically translated with a focus on being easy to understand in modern English when hearing it read. It's not a literal translation and it is more thought-for-thought than the NIV or NRSV, but it's a translation, not a paraphrase.
A Spanish Bible and Spanish audio are also available.