That didn't work. I should mention that the WiFi access point is only inches away. The problem is not the router or the wifi password.
Is the wi-fi point checked in the settings -> wifi menu?
If the phone is connected to the AP point, then it will appear checked. However, you will not get the Wifi icon in the status bar, unless the phone gets an IP address and an internet connection.
E.g. I have a secondary AP at home, connected to a wired LAN - then to an ADSL router. If I disconnect the LAN cable, then the phone will still connect to the AP, but without usable internet connection, it will continue to send data via its cellular connection.
There is a second issue with the iPhone. When the iPhone is in standby (screen off), then the wifi module in the phone is disabled. If the phone needs to send data when it is in standby (e.g. e-mail checking, push notifications, etc.) then the data can only go via cellular. [The iPhone 4 has a new power management system, which allows the wifi module to be switched on when in standby, in preference to using cellular data - but the 3GS and previous models, are stuck with cellular only in standby mode].