Yes, there are many modern roads and highways - especially in Italy - that follow the old roads. They were some seriously industrious workers - with roads throughout Europe, over mountains, across the UK and in the Middle East and Africa!
Firstly, voila, Wikipedia to the rescue!
Via Aemilia, from Rimini (Ariminum) to Placentia
Via Appia, the Appian way (312 BC), from Rome to Apulia
Via Aurelia (241 BC), from Rome to France
Via Cassia, from Rome to Tuscany
Via Flaminia (220 BC), from Rome to Rimini (Ariminum)
Via Salaria, from Rome to the Adriatic Sea (in the Marches)
Via Aemilia Scauri (109 BC)
Via Aquillia, branches off the Appia at Capua to the sea at Vibo
Via Amerina, from Rome to Ameria and Perusia
Via Canalis, from Udine, Gemona and Val Canale to Villach in Carinthia and then over Alps to Salzburg or Vienna
Via Claudia Julia Augusta (13 BC)
Via Claudia Nova (47 AD)
Via Clodia, from Rome to Tuscany forming a system with the Cassia
Via Domitiana, coast road from Naples to Formia
Via Flavia, from Trieste (Tergeste) to Dalmatia
Via Gemina, from Aquileia and Trieste through the Karst to Materija, Obrov, Lipa and Klana, from where, near Rijeka, descending towards Trsat (Tersatica) to continue along the Dalmatian coast
Via Julia Augusta (8 BC), exits Aquileia
Via Labicana, southeast from Rome, forming a system with the Praenestina
Via Ostiensis, from Rome to Ostia
Via Postumia (148 BC), from Verona across the Apennines to Genoa
Via Popilia (132 BC), two distinct roads, one from Capua to Rhegium and the other from Ariminum through the later Veneto region
Via Praenestina, from Rome to Praeneste
Via Schlavonia, from Aquileia across northern Istria to Senj and into Dalmatia
Via Severiana, Terracina to Ostia
Via Tiburtina, from Rome to Aternum
Via Traiana Nova (Italy), from Lake Bolsena to the Via Cassia. Known by archaeology only
Main article: Roman roads in Africa
Main road: from Sala Colonia to Carthage to Alexandria.
In Egypt: Via Hadriana
In Mauretania Tingitana from Tingis southward (see: Roman roads in Morocco)
Albania / Republic of Macedonia / Greece / Turkey
Via Egnatia (146 BC) connecting Dyrrhachium (on Adriatic Sea) to Byzantium via Thessaloniki
Austria / Serbia / Bulgaria / Turkey
Via Militaris (Via Diagonalis, Via Singidunum), connecting Middle Europe and Byzantium
Roman road in Cilicia in south Turkey
In France, a Roman road is called voie romaine in vernacular language.
Via Aquitania, from Narbonne, where it connected to the Via Domitia, to the Atlantic Ocean across Toulouse and Bordeaux
Via Domitia (118 BC), from Nîmes to the Pyrenees, where it joins to the Via Augusta at the Col de Panissars
Voie romaine, extending from Dunkirk to Cassel in Nord Département
Germania Inferior (Germany, Belgium, Netherlands)
Via Belgica (Boulogne-Cologne)
Lower Limes Germanicus
Interconnections between Lower Limes Germanicus and Via Belgica
Via Traiana Nova
Petra Roman Road First Century Petra, Jordan
Roman roads along the Danube
Trajan's bridge and Iron Gates road.
Via Traiana: Porolissum Napoca Potaissa Apulum road.
Via Pontica: Troesmis Piroboridava Caput Stenarum Apulum Partiscum Lugio
Romania / Bulgaria
Spain and Portugal
Iter ab Emerita Asturicam, from Sevilla to Gijón. Later known as Vía de la Plata (plata means "silver" in Spanish, but in this case it is a false cognate of an Arabic word balata), part of the fan of the Way of Saint James. Now it is the A-66 freeway.
Via Augusta, from Cádiz to the Pyrénées, where it joins to the Via Domitia at the Coll de Panissars, near La Jonquera. It passes through Valencia, Tarragona (anciently Tarraco), and Barcelona.
Camiño de Oro, ending in Ourense, capital of the Province of Ourense, passing near the village of Reboledo.
These roads connected modern Italy and Germany
Via Claudia Augusta (47) from Altinum (now Quarto d'Altino) to Augsburg via the Reschen Pass
Via Mala from Milan to Lindau via the San Bernardino Pass
Connecting Hispania and Gallia:
Ab Asturica Burdigalam
Main article: Roman roads in Britain
London-West of England Roman Roads
But clearly you can't go see them all (well you could, but it would take a while). So let's focus on the more fantastic or famous ones. Primarily, the Via Appia.
Appia teritur regina longarum viarum
"the Appian way is the queen of the long roads"
The Appian Way (Latin and Italian: Via Appia) was one of the earliest and strategically most important Roman roads of the ancient republic. It connected Rome to Brindisi, Apulia, in southeast Italy.
This is THE road. Much of it still exists today. If you know your Roman history (or watch the Starz tv show) - this is the road on which Spartacus was crucified - after his revolt, in 71 BC, 6,000 were crucified along the 200-kilometer Via Appia from Rome to Capua.
It has the Temple of Hercules, the catacombs of St Sebastian, and Mausoleum of Gallienus - among others.
Finally, while book resources starts to venture more into requesting history information rather than travel information, I can offer the following suggestions:
There's also an iPhone game loosely based on the building of the thousands of miles of road, but it's such a tenuous connection that I won't even dignify it by posting a link ;)