In my old phone I had a setting called "Area Info - cell information". When enabled it would show the location information next to the mobile network name, like:
Vodafone Foobar Street
I can't find this setting anywhere in Android :(
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Not all carriers support this. And Android does not show this by default IMHO. If you're still with the same provider and in the same area, you might want to check your app drawer. Some have an app called "Cell Info" pre-installed (I have it e.g. on my LG Optimus 4X – but in my area this feature is not supported).
Alternatively, you might want to look for some location widgets you could place on your home screen. Several weather widgets also display your current location along with the weather reports.
If the notification area would suffice, you also could take a look at GPS Notification or GPS Status & Toolbox. These apps display your current position in the notification area as long as GPS is turned on (not sure whether you can configure that to city/street, as I didn't try them out).
Another possibility would be "self-made widgets" using e.g. Zooper or UCCW. Again, I didn't try one of those, so I cannot say for sure whether they would go down to street level. But at least those widgets would go directly to your homescreen, and you can customize them heavily.