Karaite Sabbatarians must always use the calendar which dominates in the country where they live.
Those living in Israel use a combination of the State Hebrew calendar with the Julian Calendar and will have no trouble to follow the local Israeli Karaite count for the Omer. 
Those in societies where national holidays are observed on the Chinese Calendar, are strongly advised to observe Shavuot with the 5/5 Festival, Sukkot with the Autumn Moon Festival etc. so as to have no trouble getting time off work from your employer to observe them.
In countries where the Persian calendar dominates, the week of the Full Moon closest to Nowruz is observed as Hag Ha-Motzi and the rest of the year can be calculated from that using the same scheme outlined below
In most other parts of the world this Galuti calendar can be used. This week-based Calendar substitutes dates on the international calendars (Gregorian or Julian depending on what dominates where you live) for the Hebrew calendar so that even if Hebrew dates are difficult to follow at least the Holidays won't be missed by Karaite Sabbatarians in diaspora. 

01, March Aviv Rosh Chodesh is the Sunday of the 2nd week before the week of the Full Moon closest to the 21st of March.
15-21, 01 The week of the Full Moon closest to the 21st of March is observed as Hag Ha-Motzi.
22, 01 Sunday of the week after Hag Ha-Motzi is Yom Ha-Bikkurim (1st day of Omer/Paskha not Pesakh).
18, 02 Mid-Pentecost is the 4th Wednesday of the Omer/Paskha (the 25th day of Paskha).
05, 03 Sabbatarian Ascension (Thursday the 40th day of Paskha).
15, 03 Pentecost (Sunday the 50th day after Paskha)
09, 04 The Apostles' fast begins on the 2nd Monday of Tammuz
07, 05 The Sunday of the week of July 10th is the start of the three day Tzom HaHamishi for the Destruction of Solomon's Temple
10, 05 The Wednesday of the week of July 10th is the Korban Feast (All Saint's Day)
August Elul
01, 07 The Sunday of the week of September 1st is Indiction (Yom Ha-Teruah)
10, 07 The Tuesday of the 2nd week of September is Yom Kippur
15-21, 07 The Third week of September is the Feast of Tabernacles
22, 07 The Sunday of the 4th week of September is Shemnei Atzeret
8th Month Cheshvan
25, 09 The Sunday Eve to Sunday Eve Octave of November 25th is Hanukah in which the Books of Maccabees are read.
10.10 Tzom HaAsrei is the Fast of Adam from the 10th of December until Kalenda
25th of December Kalenda
1st of January Tu BiShevat all treefruit tying must be finished
6th of January Sabbatarian Epiphany
31st January Shevat is extended if necessary instead of Adar
12th Month
Adar is always the 12th month to begin Yashan ve Chadash
Purim fast (3 days)

As a point of interest, there is no one standard calendar in Karaite Judaism. Basicaly almost anything goes. Gazarians and Okbarites have a specific week based calendar where Yom Kippur always falls on a Saturday and seven days of unleavened bread follow Pesakh which (as well as Trumpets, 1st Sukkot and the 8th day) with the last day of unleavened bread always falls on a Thursday. Nevertheless, the Karaimite Calendar is not a Karaite Jewish calendar.