Amy
There are a lot really but if you're looking for one I stayed at the Attalos Hotel when I was in Athens last spring. It's clean, the staff is friendly, the breakfast is good, and best of all you have a clear view of the Acropolis from the roof top bar. It's especially brilliant if you go up at night when the entire thing is lite up. Best of all it's fairly cheap, a double room costs just about a hundred dollars (US) when they're not at peak season. If you go between late April and October, I was in the city for May day, than you're looking at just under a hundred and fifty.

The Stainless Steel Rat
Define affordable. The catch here is the Acropolis view. You pay for it. Same two rooms in a hotel will have a considerable price difference if one of them has The View:-)

Mason
The hotels view to the Acropolis itself is worth a visit, with the superbly professional staff catering to your every need.

Carmelo
go to BookMe. You can compare prices on all the major travel sites from one location. It's the only way you can guarantee the best deals for the best hotels for your money online.

Gate 1 Travel
Hector, have you looked on TripAdvisor for hotel suggestions? It’s a great site that rates hotels based on user reviews. I find it very helpful. However, I would suggest looking into The Athens Gate Hotel or The Magna Grecia Hotel. They both are said to have good views of the Acropolis. Enjoy your time in Greece!

Kalimana Paparigopoulou
Ask in your embassy first for approval !!! Official US Embassy report : The U.S. Government remains deeply concerned about the heightened threat of terrorist attacks against U.S. citizens and interests abroad. Like other countries that are members of the Schengen Agreement for free cross-border movement, Greece’s open borders with its European neighbors allow the possibility of terrorist groups entering/exiting the country with anonymity. As the first entry point into Schengen from points south and east, Greece’s long coastline and many islands also heighten the possibility that foreign-based terrorists might try to exploit Greece’s borders. Domestic terrorist organizations such as Revolutionary Struggle and “Sect of Revolutionaries” have become increasingly active against both domestic and foreign targets in Greece. Recent attacks and attempted attacks have included the use of Molotov cocktails (gasoline bombs), small arms and rifle fire, and improvised explosive devices, the largest being a 50-kilo car bomb which failed to detonate before the police arrived. The Greek police forces, other Greek governmental agencies, private American and Greek businesses, and the United States Embassy have been attacked in the last three years. Recent actions indicate that the perpetrators are indiscriminate. The risk of “being in the wrong place at the wrong time” in the event of a terrorist action is a concern for residents and visitors. U.S. tourists or residents in Greece should remain vigilant, exercise caution, and monitor local developments. Some current terrorist organizations are aligned with the ideology of the older Marxist terrorist group November 17 (N17) which targeted Greek businessmen and officials, as well as officials from NATO countries in Greece, from the mid-1970s until the early part of this decade. N17 terrorists murdered 23 people, including five U.S. Government employees. Strikes and demonstrations are a regular occurrence. Greece is a stable democracy and these activities for the most part are orderly and lawful. However, a wave of incidents started when a teenager was shot and killed in an encounter with the police in December 2008. Incidents occurred throughout Greece, but the primary sources of violence were in Athens and Thessaloniki, Protestors there engaged in violent confrontations with the police and carried out destructive vandalism and rioting in localized areas, some of which are areas frequented by tourists, injuring numerous police officers. Riot control procedures often include the use of tear gas. Visitors should keep abreast of news about demonstrations from local news sources and hotel security. When there are demonstrations, visitors should be aware of and avoid places where demonstrators frequently congregate, such as the Polytechnic University area, Exarchia, Omonia, and Syntagma Squares in Athens, and Aristotle Square in Thessaloniki. The Omonia and Exharchia areas of Athens are at particular risk for crime and politically-motivated violence; U.S. Embassy personnel and their families have been urged strongly to avoid these areas between 9 pm and dawn.