What is the English word for a regular city street that happens to be situated along a river?
(For the record, the Russian word is набережная, but I can't seem to find the appropriate translation.)
You can find these streets referred to as river roads, often with River Road or River Street being part of their proper names. There are many books that use the term, including The Majesty of the River Road, where the introduction begins
Along the road that follows the course of the Mississippi stand many majestic structures that have withstood the test of time.
Here a picture where it's part of the proper name, Savannah, Georgia's (USA) River Street (ref)
While embankment might describe a bank, mound, dike, levee, or the like, and is typically raised to hold back water, it may carry a roadway or railway, but not necessarily (ref):
London has the roads called Victoria Embankment and the Albert Embankment on opposite sides of the river; it also has the Strand — strand as a noun refers to a beach.
“The Embankment” generally means the path along the south side of the Thames between Westminster Bridge and Lambeth Bridge, opposite the Palace of Westminster.
As the Wikipedia article explains, Embankment does mean an earthwork built to contain a river or support a railway or road; but (as the article notes) can also refer to a roadway atop such a structure. Usually, such a use is qualified as in Victoria/Albert/Chelsea Embankment.
The commentators have already given viable suggestions. I'd like to add towpath, which has the exact relationship you describe - a street or path that runs alongside a river or canal, which horses used to walk along, pulling or towing barges behind them, hence the name.
According to this source, you can also use 'towing path', although that's less common in my experience.
There is no word that means exactly and unambiguously “street that happens to be along a river bank”, but there are several words that can be used to refer to a river bank where pedestrians or vehicles can tread (not necessarily both). Which word is used in a given place is a matter of regional variation and historical happenstance. These words have an etymological connotation that may or may not apply in any given use.
A quay, wharf, dock or pier is an area along a river or sea side where boats can load and unload cargo. A quay or dock is artificial, a wharf can be either artificial or natural. These words can refer to the wooden or stone (or earth, for a quay) structure where boats moor, or to the road that runs alongside the shore, or to a larger area going up to the whole neighborhood (as in Canary Wharf in the London Docklands). Among these, I think quay is the one most commonly used as a street name. This list of names features example of all these terms.
An embankment is etymologically a levee, a raised band of earth on the side of the river which can be natural or artificial. Usually pedestrians or vehicles can move on the embankment or on a road next to it. By extension, any river-side road can be called an embankment. The London Embankment is a famous example; several streets bear that name (Victoria Embankment, Chelsea Embankment).
Regarding some other words that have been mentioned in this thread, a terrace is a raised bank of earth (among other meanings). An esplanade is a wide space or street. Both terms can be used for a waterfront street, but they do not imply that the street is near water, so they are best avoided as a translation. psmears also suggests corniche, a road at the top of a cliff, often but not always along a river or sea shore.
You can also refer to such a street using a qualifier: it is simply a riverside street, or a river street or river road.
If there is no local usage to fall back on, I would go with embankment for a riverside road in the countryside, or a quay in a city, especially but not exclusively if boats can moor.