Due to the length of this post, I'll break it up by city pairs – Johannesburg, South Africa and Dakar, Senegal then Capetown, South Africa and Accra, Ghana. Though there are lie-flats available to Lagos, Nigeria (LOS), I will be omitting that section since the US Department of State considers criminal threats in the area "critical".
Methodology for this article: I first checked Flight Stats for Departures and Arrivals at these particular airports to gauge what my options were. Then I matched the results with Seat Guru to see if I'd get a lie-flat option. From there, I had to figure out how to get there using various awards. Here is what I found.
To Johannesburg, South Africa (JNB)
Fly American from North American city to London (LHR) then continue with British Airways from LHR to JNB.
BA's First and Club World Class offer lie-flat seats. It costs 100,000 miles one-way for First MileSAAver Awards and 75,000 miles one-way for Business/First Mile SAAver Awards using AA miles.
I checked on the British Airways (BA) and Japan Airlines (JAL) sites as well as calling AA for availability but confirmed that LHR to JNB had sparse availability in First and Business. There aren't many departure dates though April 2014 had a few worth looking at. These awards charge a high surcharge so they weren't very desirable to begin with. Book as far out as possible.
Fly Cathay Pacific from North American city to Hong Kong (HKG) then continue HKG to JNB.
Cathay Pacific's (CX) Business class offers lie-flat seats. None of Cathay's 747s are getting the new business class however. There was miscellaneous business class availability for HKG-JNB from Feb-Mar as confirmed by JAL and the AA call-center. Though Cathay Pacific flies a First class route from LAX-HKG there is only Business class available for HKG-JNB.
To book this award requires 235,000 in First (LAX-HKG)/Business (HKG-JNB) and 210,000 in Business using AA miles. AA requires you fly a transatlantic route to get to Africa so you must book this as 2 separate awards if you want to fly CX. Another option would be using Alaska Mileage Plan for 140,000 round-trip for First and 125,000 round-trip for Business. Book via Alaska Airlines Reservations at 1-800-252-7522 (TTY/TDD: 1-800-682-2221) , daily between 5:00 a.m. and midnight (PT).
Fly Delta from Atlanta (ATL) to JNB.
Business Elite Class offers lie-flat seats. There was horrible availability with no low priced awards available. From now until April 2014, only 5 scattered days from ATL were available. It costs 120,000 Skymiles round-trip for Business Elite Class.
Fly Air France from North American city to Paris (CDG) then continue CDG to JNB.
Air France La Premiére (First) offers closed suites. La Premiére cannot be booked with DL Skymiles. It only releases first class award availability to Flying Blue elite members. If you are FB elite or know one who will help book an award for you, you can proceed with this route. Otherwise, it's best to wait for another status match opportunity. I found lots of availability for this award. It costs 200,000 Flying Blue miles one-way from LAX-CDG-JNB, for example, with ~$300 or so taxes and surcharges.
Fly South African Airways from JFK or Washington (IAD) to JNB.
South African Airways' (SAA) Business class offers a flat bed option. I couldn't really find much availability with the non-stop route. I found a few scattered Biz Saver with mixed cabins from Nov-Jan but connecting through LHR. UA agent said it may be too far in advance and the carrier hasn't decided how many seats it wants to release to partners. If it does show up, it costs 120,000 miles at the Saver level. This is also bookable with US Airways miles which requires 110,000 round-trip.
Fly Virgin Atlantic from North American city to LHR then continue LHR to JNB.
Upper Class has fully flat beds. It costs 220,000 using Virgin Atlantic miles to get from LAX to JNB. I looked this up on virgin-atlantic.com's spending calculator and called their call-center to reconfirm the pricing. The surcharges for the last segment alone is between 1700 to 2000 USD. Mile Value however documents a way to possibly get around this.
Update: To make a better, more comprehensive list of your lie-flat options to Africa, it's best to point out that Turkish Airlines also offers a lie-flat in Business from Istanbul (IST) to JNB, Swiss offers suites in First and lie-flats in Business from Zurich (ZRH) to JNB or you can get creative and use any mixture of SAA/UA/LH to get to your destination. (HT to Seth)
To Dakar, Senegal (DKR)
Fly South African Airways from IAD to DKR.
Business Class offers a flat-bed option. I couldn't find availability using ua.com but if it's there it also costs 120,000 UA miles at the Saver option or 110,000 US Airways miles round-trip.
Fly Brussels Airlines from North American City to Brussles (BRU) then continue BRU to DKR.
I only happened on this route by chance on United.com looking for SAA availability but instead found excellent availability with Brussels Airlines from IAD to DKR in Biz Saver. Though the website doesn't confirm lie-flat seats available, Seat Guru and the Brussels Airlines' call center confirms its A330 has lie-flat seats. 120,000 UA or 110,000 US Airways miles round-trip are required.
Conclusion: British Airways and Virgin Atlantic surcharges are rather unattractive and compared to them, Air France's ~$600 or so doesn't seem all that bad. Delta's notoriously non-existent low level award pricing again prevails in the above example. Star Alliance seems to have the better options in terms of saver level redemptions and availability. Cathay Pacific is a fun option to maximize your time in the air even if you are on the so-called "coffin-class" on one portion of your journey.